File size: 3,556 Bytes
ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import Mathlib
import OAI.Computability.MaxCut.Games.LabelBound
namespace OAI
/-! Exact interfaces between the information vectors and finite game laws. -/
namespace MaxCutGames.Foundations.Information
open scoped BigOperators
variable {α β : Type*} [Fintype α] [Fintype β]
theorem gameLaw_isProbability (μ : Games.FiniteDistribution α) : IsProbability μ.weight :=
⟨μ.nonnegative, μ.normalized⟩
def toGameLaw (p : α → ℝ) (hp : IsProbability p) : Games.FiniteDistribution α where
weight := p
nonnegative := hp.1
normalized := hp.2
def gameLawEquiv : Games.FiniteDistribution α ≃ {p : α → ℝ // IsProbability p} where
toFun μ := ⟨μ.weight, gameLaw_isProbability μ⟩
invFun p := toGameLaw p.1 p.2
left_inv μ := by cases μ; rfl
right_inv p := by cases p; rfl
theorem totalVariation_gameLaw (μ ν : Games.FiniteDistribution α) :
totalVariation μ.weight ν.weight = μ.totalVariation ν := rfl
noncomputable def gameConditionalKernel (μ : Games.FiniteDistribution (α × β))
(fallback : Games.FiniteDistribution β) (a : α) : Games.FiniteDistribution β :=
toGameLaw (conditionalKernel μ.weight fallback.weight a)
(conditionalKernel_isProbability μ.weight fallback.weight
(gameLaw_isProbability μ) (gameLaw_isProbability fallback) a)
theorem gameConditionalKernel_recombine (μ : Games.FiniteDistribution (α × β))
(fallback : Games.FiniteDistribution β) (a : α) (b : β) :
firstMarginal μ.weight a * (gameConditionalKernel μ fallback a).weight b =
μ.weight (a, b) :=
marginal_mul_conditionalKernel μ.weight fallback.weight (gameLaw_isProbability μ) a b
noncomputable def gameConditionalProduct (μ : Games.FiniteDistribution (α × β))
(fallback : Games.FiniteDistribution β) (newInput : Games.FiniteDistribution α) :
Games.FiniteDistribution (α × β) :=
toGameLaw (fun ab => newInput.weight ab.1 * conditionalKernel μ.weight fallback.weight ab.1 ab.2)
(alternativeInput_conditionalProduct_isProbability μ.weight fallback.weight newInput.weight
(gameLaw_isProbability μ) (gameLaw_isProbability fallback) (gameLaw_isProbability newInput))
theorem gameConditionalProduct_weight (μ : Games.FiniteDistribution (α × β))
(fallback : Games.FiniteDistribution β) (newInput : Games.FiniteDistribution α)
(a : α) (b : β) :
(gameConditionalProduct μ fallback newInput).weight (a, b) =
newInput.weight a * (gameConditionalKernel μ fallback a).weight b := rfl
theorem gameCondition_weight (μ : Games.FiniteDistribution α) (event : α → Bool)
(positive : 0 < μ.probability event) :
(μ.condition event positive).weight =
posterior μ.weight (fun a => if event a then 1 else 0) (μ.probability event) := by
funext a
cases he : event a <;> simp [Games.FiniteDistribution.condition, posterior, he]
theorem gameCondition_relativeEntropy_le (μ : Games.FiniteDistribution α) (event : α → Bool)
(positive : 0 < μ.probability event) :
relativeEntropy (μ.condition event positive).weight μ.weight ≤
Real.log (1 / μ.probability event) := by
rw [gameCondition_weight]
apply posterior_relativeEntropy_le μ.weight (fun a => if event a then 1 else 0)
(gameLaw_isProbability μ)
· intro a
cases event a <;> norm_num
· intro a
cases event a <;> norm_num
· exact positive
· unfold Games.FiniteDistribution.probability
apply Finset.sum_congr rfl
intro a _
cases event a <;> simp
end MaxCutGames.Foundations.Information
end OAI
|